Euclidean correlators at imaginary spatial momentum and their relation to the thermal photon emission rate

The photon emission rate of a thermally equilibrated system is determined by the imaginary part of the in-medium retarded correlator of the electromagnetic current transverse to the spatial momentum of the photon. In a Lorentz-covariant theory, this correlator can be parametrized by a scalar function ${\cal G}_R(u\cdot {\cal K},{\cal K}^2)$, where $u$ is the fluid four-velocity and ${\cal K}$ corresponds to the momentum of the photon. We propose to compute the analytic continuation of ${\cal G}_R(u\cdot {\cal K},{\cal K}^2)$ at fixed, vanishing virtuality ${\cal K}^2$, to imaginary values of the first argument, $u\cdot {\cal K}= i\omega_n$. At these kinematics, the retarded correlator is equal to the Euclidean correlator $G_E(\omega_n, k=i\omega_n)$, whose first argument is the Matsubara frequency and the second is the spatial momentum. The Euclidean correlator, which is directly accessible in lattice QCD simulations, must be given an imaginary spatial momentum in order to realize the photon on-shell condition. Via a once-subtracted dispersion relation that we derive in a standard way at fixed ${\cal K}^2=0$, the Euclidean correlator with imaginary spatial momentum is related to the photon emission rate. The relation allows for a more direct probing of the real-photon emission rate of the quark-gluon plasma in lattice QCD than the dispersion relations which have been used so far, the latter being at fixed spatial photon momentum $k$ and thus involving all possible virtualities of the photon.